With the anglosphere mainstream being either Protestant or Secular, it’s been quite easy for them to use the word more metaphorically these days.

Remind me Chrysovalanti: is the Greek word you’d use προσκύνημα? Because pilgrimage is certainly one of its meanings, but it applies rather more broadly than pilgrimage: it applies to any veneration of relics or saints, and does not require that you travel far. Of course, the veneration of relics is no more Protestant or Secular than a pilgrimage is.